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Theorem tppreqb 4079
Description: An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)
Assertion
Ref Expression
tppreqb  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  <->  { A ,  B ,  C }  =  { A ,  B } )

Proof of Theorem tppreqb
StepHypRef Expression
1 3ianor 999 . . . 4  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  <->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
2 df-3or 983 . . . 4  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
)  <->  ( ( -.  C  e.  _V  \/  -.  C  =/=  A
)  \/  -.  C  =/=  B ) )
31, 2bitri 252 . . 3  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  <->  ( ( -.  C  e.  _V  \/  -.  C  =/=  A
)  \/  -.  C  =/=  B ) )
4 orass 526 . . . . 5  |-  ( ( ( ( -.  C  e.  _V  \/  -.  C  =/=  A )  \/  -.  C  =/=  B )  \/ 
-.  C  e.  _V ) 
<->  ( ( -.  C  e.  _V  \/  -.  C  =/=  A )  \/  ( -.  C  =/=  B  \/  -.  C  e.  _V ) ) )
5 ianor 490 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  <-> 
( -.  C  e. 
_V  \/  -.  C  =/=  A ) )
6 tpprceq3 4078 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  { B ,  A ,  C }  =  { B ,  A } )
75, 6sylbir 216 . . . . . . 7  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  A
)  ->  { B ,  A ,  C }  =  { B ,  A } )
8 tpcoma 4034 . . . . . . 7  |-  { B ,  A ,  C }  =  { A ,  B ,  C }
9 prcom 4016 . . . . . . 7  |-  { B ,  A }  =  { A ,  B }
107, 8, 93eqtr3g 2480 . . . . . 6  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  A
)  ->  { A ,  B ,  C }  =  { A ,  B } )
11 orcom 388 . . . . . . . 8  |-  ( ( -.  C  =/=  B  \/  -.  C  e.  _V ) 
<->  ( -.  C  e. 
_V  \/  -.  C  =/=  B ) )
12 ianor 490 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  <-> 
( -.  C  e. 
_V  \/  -.  C  =/=  B ) )
1311, 12bitr4i 255 . . . . . . 7  |-  ( ( -.  C  =/=  B  \/  -.  C  e.  _V ) 
<->  -.  ( C  e. 
_V  /\  C  =/=  B ) )
14 tpprceq3 4078 . . . . . . 7  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B } )
1513, 14sylbi 198 . . . . . 6  |-  ( ( -.  C  =/=  B  \/  -.  C  e.  _V )  ->  { A ,  B ,  C }  =  { A ,  B } )
1610, 15jaoi 380 . . . . 5  |-  ( ( ( -.  C  e. 
_V  \/  -.  C  =/=  A )  \/  ( -.  C  =/=  B  \/  -.  C  e.  _V ) )  ->  { A ,  B ,  C }  =  { A ,  B } )
174, 16sylbi 198 . . . 4  |-  ( ( ( ( -.  C  e.  _V  \/  -.  C  =/=  A )  \/  -.  C  =/=  B )  \/ 
-.  C  e.  _V )  ->  { A ,  B ,  C }  =  { A ,  B } )
1817orcs 395 . . 3  |-  ( ( ( -.  C  e. 
_V  \/  -.  C  =/=  A )  \/  -.  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B }
)
193, 18sylbi 198 . 2  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B }
)
20 df-tp 3941 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
2120eqeq1i 2428 . . 3  |-  ( { A ,  B ,  C }  =  { A ,  B }  <->  ( { A ,  B }  u.  { C } )  =  { A ,  B }
)
22 ssequn2 3577 . . . 4  |-  ( { C }  C_  { A ,  B }  <->  ( { A ,  B }  u.  { C } )  =  { A ,  B } )
23 snssg 4071 . . . . . . 7  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  { C }  C_  { A ,  B } ) )
24 elpri 3953 . . . . . . . 8  |-  ( C  e.  { A ,  B }  ->  ( C  =  A  \/  C  =  B ) )
25 nne 2600 . . . . . . . . . 10  |-  ( -.  C  =/=  A  <->  C  =  A )
26 3mix2 1175 . . . . . . . . . 10  |-  ( -.  C  =/=  A  -> 
( -.  C  e. 
_V  \/  -.  C  =/=  A  \/  -.  C  =/=  B ) )
2725, 26sylbir 216 . . . . . . . . 9  |-  ( C  =  A  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
28 nne 2600 . . . . . . . . . 10  |-  ( -.  C  =/=  B  <->  C  =  B )
29 3mix3 1176 . . . . . . . . . 10  |-  ( -.  C  =/=  B  -> 
( -.  C  e. 
_V  \/  -.  C  =/=  A  \/  -.  C  =/=  B ) )
3028, 29sylbir 216 . . . . . . . . 9  |-  ( C  =  B  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
3127, 30jaoi 380 . . . . . . . 8  |-  ( ( C  =  A  \/  C  =  B )  ->  ( -.  C  e. 
_V  \/  -.  C  =/=  A  \/  -.  C  =/=  B ) )
3224, 31syl 17 . . . . . . 7  |-  ( C  e.  { A ,  B }  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
3323, 32syl6bir 232 . . . . . 6  |-  ( C  e.  _V  ->  ( { C }  C_  { A ,  B }  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) ) )
34 3mix1 1174 . . . . . . 7  |-  ( -.  C  e.  _V  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
3534a1d 26 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( { C }  C_  { A ,  B }  ->  ( -.  C  e. 
_V  \/  -.  C  =/=  A  \/  -.  C  =/=  B ) ) )
3633, 35pm2.61i 167 . . . . 5  |-  ( { C }  C_  { A ,  B }  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
3736, 1sylibr 215 . . . 4  |-  ( { C }  C_  { A ,  B }  ->  -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B ) )
3822, 37sylbir 216 . . 3  |-  ( ( { A ,  B }  u.  { C } )  =  { A ,  B }  ->  -.  ( C  e. 
_V  /\  C  =/=  A  /\  C  =/=  B
) )
3921, 38sylbi 198 . 2  |-  ( { A ,  B ,  C }  =  { A ,  B }  ->  -.  ( C  e. 
_V  /\  C  =/=  A  /\  C  =/=  B
) )
4019, 39impbii 190 1  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  <->  { A ,  B ,  C }  =  { A ,  B } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2594   _Vcvv 3017    u. cun 3372    C_ wss 3374   {csn 3936   {cpr 3938   {ctp 3940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-v 3019  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-sn 3937  df-pr 3939  df-tp 3941
This theorem is referenced by: (None)
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